Equivalence of two Bochkov-Kuzovlev equalities in quantum two-level systems

Abstract

We present two kinds of Bochkov-Kuzovlev work equalities in a two-level system that is described by a quantum Markovian master equation. One is based on multiple time correlation functions and the other is based on the quantum trajectory viewpoint. We show that these two equalities are indeed equivalent. Importantly, this equivalence provides us a way to calculate the probability density function of the quantum work by solving the evolution equation for its characteristic function. We use a numerical model to verify these results.PACS numbers: 05.70. Ln, Introduction. In the past decade, extending classical fluctuation relations [1][2][3][4][5][6][7][8][9][10][11][12][13] into nonequilibrium quantum regime is attracting intensive interest [1,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. In the literature, the quantum measurement [15,17,20,21,27] and the quantum trajectory in Hilbert space [19,24,[28][29][30][33][34][35][36] are two widely used fundamental concepts. As an alternative to those two concepts, very recently, Chetrite and Mallick [37], and we [38] showed that, in the isolated Hamiltonian system and some quantum Markovian master equations (QMME) [34,46], the quantum work equalities [1,6] can be derived based on the time-reversal and quantum Feynman-Kac formulas. In contrast with the conventional work equalities written as statistical average of exponential functions [19,24,[28][29][30], which we name them the c-number equalities, the newly obtained equalities that are named the q-number equalities are remarkable analogies with their classical counterparts [37,38]. Even so, in the case of the QMMEs, the exact relationship of the q-number and c-number equalities, and whether the q-number equalities are useful besides their forms have not been seriously considered. In this Rapid communication, we use a driven quantum two-level system (TLS) with dissipation to prove that, the q-number and the c-number Bochkov-Kuzovlev equalities (BKE) are indeed equivalent for a specific class of QMMEs. An important consequence of this investigation is that we find an efficient way to calculate the probability density function (pdf) of the quantum work for these systems without doing the quantum jump simulation [34,36].

Nonequilibrium potential and fluctuation theorems for quantum maps

“…with a corresponding integral fluctuation theorem that follows readily, like in (17). Thus, for a concatenation of maps implemented in sequence, we merely have to add the changes in the nonequilibrium potential along the trajectory.…”

“…General quantum Markov semigroups were explored by Crooks using time-reversed or dual maps [14], which were then applied by Horowitz et al to nonequilibrium quantum jump trajectories [5,15]. An alternative, operator formulation for driven Lindbald master equations was independently developed by Chetrite and Mallick [16], and its equivalence to the quantum jump approach was investigated by Liu [17,18]. Fluctuation theorems under unital CPTP maps for thermodynamic quantities, like work, energy, and information-theoretic entropy, have appeared in numerous works [19][20][21], while predictions for nonunital CPTP maps usually take the form of an integral fluctuation theorem with a so-called correction [6,[20][21][22][23].…”

Quantum corrections of work statistics in closed quantum systems

“…In addition, if there are many identified particles in quantum systems, quantum statistics have to be taken into account. Finally, for open quantum systems, we have established the quantum FK formula as well [4,41,42]. The exact meaning of quantum-classical correspondence in these situations is worth investigating in detail.…”

“…Substituting all relevant quantities into Eqs. (40) and (41) and after some algebraic calculations, we obtain the two corrections as follows:…”

Fluctuations of work in nearly adiabatically driven open quantum systems

“…To model the work statistics, the quantum jump method can be used, as previously studied for the Lindblad equation with a time-independent dissipative part [29,[33][34][35]. However, in the presence of a strong drive, the dissipative part may become time dependent, which has been proven to affect the fluctuation relations [23,[38][39][40].…”